No, you make excellent points.
The total picture is indeed very muddied.
I'll say.
I would imagine the Saudis had to make a request for the US assistance, and at the time, the interests of all Sunni-Arab nations had aligned with those of Israel, in a way they never had before the Iran nuke negotiations focussed everyone's mind.
Is it possible there was a quid-pro-quo, and it involved a threat by the Sunnis to request Bibi's help in Yemen if Obama didn't help them? Israel, in my opinion, would have jumped on the opportunity. The Saudis considered Iranian influence in Yemen to be an existential problem, and the coalition was put together in a matter of days, after years of disfunction within the Arab League from which it sprang.
In other words, Bibi, you help us with Yemen, and we'll provide you with airspace and logistics to attack Iran?
And it never happened because Obama caved, and if you'll notice, so did Iran. In Yemen at least. Maybe Obama told the Iranians, and they considered it to be the lesser evil to give up on Yemen, because the Mullahs REALLY don't want the "Arab Coalition plus Israel" to become a reality.